Finite Group Databasebeta

← Prev (#72343978) Next (#72343980) →

#72343979

order3072
automorphism?
homologyAbelianization (H1) = C23 · Schur multiplier (H2) = C25

Aliases

No aliases for this group.

Splittings

Fitting#392665 ⋊ D3

Subgroups

subgroupG/H
center, socleC22#11713272
commutator, cosocleC2×#5950C23
FrattiniC23×C4C22×S4
Fitting#392665D3
hypercenterC2×C8#1020⋊C2
hypocenter#997#120
LCSG ▷ C2×#5950 ▷ #997
UCSC1 ◁ C22 ◁ C2×C4 ◁ C2×C8

3838 conjugacy classes of subgroups; only the 57 normal subgroups are shown.

normal subgroups only

This group has 3838 conjugacy classes of subgroups — too many to draw. Only its 57 normal subgroups are shown.

Dotted edges: containments with no intermediate normal subgroup — omitted non-normal subgroups may lie between.

Full subgroup data →

Presentation

minimal degreeμ(G) = 36
transitive degreeμt(G) = 64

A minimal faithful permutation representation — 3 generators on 36 points:

(2,4)(5,28,23)(6,19,22)(7,35,33)(8,21)(9,10,20)(11,34,27)(12,29,25)(13,30)(14,15,31)(16,36,18)(17,32)(24,26);(6,11)(7,16)(8,20)(12,15)(13,29)(14,25)(17,35)(18,33)(19,26)(21,28)(22,27)(24,31)(30,34)(32,36);(1,2)(3,4)(5,6)(7,11)(8,19)(9,22)(10,15)(12,16)(13,28)(14,23)(17,34)(18,27)(20,30)(21,31)(24,35)(25,33)(26,36)(29,32)

Polycyclic presentation:

2^5,3,2^5|e^2=k,b^a=be,c^a=cd,e^a=ek,e^b=ek,f^a=f^2,f^b=f^2,g^a=ghik,g^b=gh,g^c=gijk,g^f=h,h^a=hjk,h^c=hjk,h^f=gh,i^a=ij,i^b=ij,i^f=ij,i^g=ik,i^h=ik,j^f=i,j^g=jk

Conjugacy classes & character table

73 conjugacy classes — view the full character table.

Relationships

maximal subgroup of#72343985, #72344011
maximal normal subgroup of#72343985, #72344011

GAP

Past the SmallGroups library coverage — no SmallGroups id. Construct it from its polycyclic presentation:

gap> F := FreeGroup(IsSyllableWordsFamily, "a", "b", "c", "d", "e", "f", "g", "h", "i", "j", "k");;
gap> coll := SingleCollector(F, [2, 2, 2, 2, 2, 3, 2, 2, 2, 2, 2]);;
gap> SetPower(coll, 5, F.11);;
gap> SetConjugate(coll, 2, 1, F.2*F.5);;
gap> SetConjugate(coll, 3, 1, F.3*F.4);;
gap> SetConjugate(coll, 5, 1, F.5*F.11);;
gap> SetConjugate(coll, 5, 2, F.5*F.11);;
gap> SetConjugate(coll, 6, 1, F.6^2);;
gap> SetConjugate(coll, 6, 2, F.6^2);;
gap> SetConjugate(coll, 7, 1, F.7*F.8*F.9*F.11);;
gap> SetConjugate(coll, 7, 2, F.7*F.8);;
gap> SetConjugate(coll, 7, 3, F.7*F.9*F.10*F.11);;
gap> SetConjugate(coll, 7, 6, F.8);;
gap> SetConjugate(coll, 8, 1, F.8*F.10*F.11);;
gap> SetConjugate(coll, 8, 3, F.8*F.10*F.11);;
gap> SetConjugate(coll, 8, 6, F.7*F.8);;
gap> SetConjugate(coll, 9, 1, F.9*F.10);;
gap> SetConjugate(coll, 9, 2, F.9*F.10);;
gap> SetConjugate(coll, 9, 6, F.9*F.10);;
gap> SetConjugate(coll, 9, 7, F.9*F.11);;
gap> SetConjugate(coll, 9, 8, F.9*F.11);;
gap> SetConjugate(coll, 10, 6, F.9);;
gap> SetConjugate(coll, 10, 7, F.10*F.11);;
gap> UpdatePolycyclicCollector(coll);;
gap> G := GroupByRws(coll);